CONVERGENCE PROPERTIES OF THE PARAMETER VECTOR IN REAL-TIME IDENTIFICATION OF LINEAR SYSTEMS

摘要

The convergence properties of the parameter vector in a real¬time identification process of linear systems are investigated. The system is identified in terms of the coefficients of its differential equation. The main reason for this choice is that unique and globally asymptotically stable solutions can be obtain¬ed. If all the components of the parameter vector converge uniquely from some initial set of values to their steady-state in a finite interval of time, the system is completely identifi¬able. Employing a statistical approach the ensemble averaged con¬vergence process is expressed in terms of the eigenvalues of a covariance matrix which contains all the data regarding input, order and structure of the system, type of criterion function and descent law. The principal factors which determine the parameter noise vector are analyzed. The concept of criterion surface elongation which is introduced, serves as an effective figure of merit from which the convergence process can be predicted. It is shown that almost always the elongation, which is the ratio between the largest and the smallest eigenvalue increases rapidly with the order of the system. The principal axes of the multi-dimensional ellipsoid describing the criterion surface are, in general, not .aligned with the coordinate axes of the parameter space. Consequently all parameters converge poorly and the time required for complete identification may become excessively large. Numerical examples of typical systems are given. If, however, the numerator and denominator polynomials are factored, the system is eventually identified in terms of its poles and zeros. It is demonstrated that by this operation, the dominant modes are separated from the rest and their values converge to their steady state very rapidly. Thus, while maintaining the advantages of uniqueness and stability provided by the original parameter space, factorization provides the opportunity to trade in some of the less important modes against the time required for the identification of the dominant modes. It is also shown that the frequency response of the system can be identified within the time interval required for the parameter vector to converge to a "zero-error" subspace. This interval is much shorter than the one needed for convergence to the origin of the parameter space.